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Question

I am having difficulties with this question:

The approximate inside diamete of the aorta is 0.51 cm; that of a capillary is 8 microm. The approximate average blood flow speed is 1.2 m/s in the aorta and 1.2 cm/s in the capillaries. If all the blood in the aorta eventually flows through the capillaries, estimate the number of capillaries in the circulatory system.

Answer 1

assume volumetric flowrate is constant (ie total liters / sec)

so Volumetric flowrate = linear flowrate x cross sectional area

for aorta, lets call

Volumetric flowrate = Vfa
linear flowrate= Lfa
cross sectional area = Csaa = pi x da^2 / 4
(where da is diameter of aorta)

So Vfa = Lfa x pi x da^2 / 4

for capillaries,

volumetric flowrate = Vfc = number of capillaries x linear flowrate x cross sectional area = n x Lfc x Csac
(where n = number of capillaries)
and let Csac = pi x dc^2/ 4
(where dc = diameter of capillary)

so that

Vfc = n x Lfc x pi x dc^2 / 4

again, assume Vfa = Vfc so that

Lfa x pi x da^2 / 4 = n x Lfc x pi x dc^2 / 4

simplifying....

[(Lfa) x (da^2)] / [(Lfc) x (dc^2)] = n

watch units.... ready?

n = [ (1.2 m/s) x (.51 cm x 1 m/100 cm)^2] / [ (1.2 cm/s x 1 m / 100cm) x (8 microns x 1m/10^6 microns)]
= [3.1212 x 10^-5 m^3/s] / [ 7.68 x 10^-13 m^3/s]

= 4.06 x 10^7 capillaries.

Answer 2

You need to determine the volumetric flow rate in the aorta as well as the volumetric flow rate in a single capillary. Since the total volumetric flow rate is the same ("all the blood in the aorta eventually flows through the capillaries"), you need to divide the aorta volumetric flow rate by the capiallary volumetric flow rate to get the number of capillaries.

Volumetric flow rate is equal to the cross-sectional area times the velocity of flow, and the cross-sectional area of a circle is equal to π(d^2)/4, where d is the diameter. Volumetric flow rate can be written as V' (normally V with a dot over it, but the prime mark is serviceable here), so you have V' = Av = π(d^2)v/4, where A is cross-sectional area and v is velocity. I'd suggest rewriting your diameters in terms of meters. 0.51 cm = 0.0051 m and 8 um = 8 x 10^-6 m. Then use the formula for volumetric flow rate to find V'_aorta and V'_capillary. The number of capillaries will be N = (V'_aorta) / (V'_capillary).